G3 - Angles in triangles and polygons

G3 - Angles in triangles and polygons

Derive the angle sum of a triangle, then build angle sums for polygons from triangles. Use interior and exterior angles to work out missing angles and identify a regular polygon from its corners.

Why triangle angles total 180 degrees

A regular polygon has an interior angle of 156∘156^\circ. Can you discover how many sides it has without seeing the shape? A small outside turn will reveal the answer. Start with the triangle, which supplies every polygon's angle sum.

You already know

Angles on a straight line total 180∘180^\circ. Alternate angles on parallel lines are equal. A regular polygon has equal sides and equal interior angles.

Take any triangle ABCABC. Draw a straight line through CC, parallel to ABAB. The angles at AA and BB can be copied beside the angle at CC, using alternate angles. All three now fit on one straight line.

[DIAGRAM: asset_name: Deriving the triangle angle sum; asset_slug: aqa8300_g03_triangle_sum; caption: Alternate angles place a, c and b together on a straight line through C; description: Triangle ABC has base angles a and b and top angle c. A line through C parallel to AB creates alternate angles a and b beside c. Their sum is 180 degrees.]
Alternate angles place a, c and b together on a straight line through C

Triangle angle sum

For any triangle in a plane,

A+B+C=180∘A+B+C=180^\circ

where AA, BB and CC are its three interior angles.
Not on the formulae sheet: learn it.

The derivation does not use any equal sides, so the result works for a thin, tilted or obtuse triangle as well as a familiar upright one.

Worked example 1: a missing angle, non-calculator

Question. Two angles in a triangle are 47∘47^\circ and 68∘68^\circ. Work out the third angle.

Plan. Add the known angles, then subtract from 180∘180^\circ.

47+68=115known anglesx=180−115angles in a triangle=65subtract\begin{aligned} 47+68&=115 &&\text{known angles}\\ x&=180-115 &&\text{angles in a triangle}\\ &=65 &&\text{subtract} \end{aligned}

Why this step? Three sides make one triangle, so its interior-angle total is 180∘180^\circ.

Answer. 65∘65^\circ.

Check. 47+68+65=18047+68+65=180.

The second example changes one known angle to a right angle. You can subtract that first to simplify the calculation.

Worked example 2: a right-angled triangle, non-calculator

Question. A right-angled triangle has one other angle of 34∘34^\circ. Work out the remaining angle.

Plan. Remove the right angle, then the known acute angle.

180−90=90total left for the acute anglesx=90−34remove the known acute angle=56subtract\begin{aligned} 180-90&=90 &&\text{total left for the acute angles}\\ x&=90-34 &&\text{remove the known acute angle}\\ &=56 &&\text{subtract} \end{aligned}

Why this step? The two non-right angles of a right-angled triangle must share 90∘90^\circ.

Answer. 56∘56^\circ.

Check. 90+34+56=18090+34+56=180, and both remaining angles are acute.

A triangle cannot have two right angles or two obtuse angles: those would already use at least the entire total, leaving no positive third angle.

Watch out

Wrong: use 360∘360^\circ because a triangle is a closed shape. Right: its interior angles sum to 180∘180^\circ. A full turn is a different quantity.

The same subtraction works whenever the triangle has enough stated angle information.

Build a polygon from triangles

A quadrilateral can be split into two triangles by one diagonal, giving an angle sum of 2×180∘=360∘2\times180^\circ=360^\circ. A pentagon splits into three triangles, so its angle sum is 540∘540^\circ.

Draw diagonals from one vertex of a convex polygon. Each new triangle uses one more side, and the total number of triangles is two fewer than the number of sides.

[DIAGRAM: asset_name: Pentagon triangulation; asset_slug: aqa8300_g03_polygon_sum; caption: Three triangles fill a pentagon without adding any new angles inside it; description: Pentagon ABCDE is divided by AC and AD into three numbered triangles, all sharing vertex A. Five sides give three triangles.]
Three triangles fill a pentagon without adding any new angles inside it

Convex polygon

A polygon with every interior angle less than 180∘180^\circ. A regular pentagon is convex. A polygon with an inward-pointing corner has a reflex interior angle and is concave.

The diagonals inside a convex polygon give a direct derivation. The same angle-sum formula also holds for a simple concave polygon, although you must choose a triangulation that stays inside it. A simple polygon has no crossing sides.

Interior-angle sum

For a simple polygon with nn sides,

S=(n−2)×180∘S=(n-2)\times180^\circ

where nn is a whole number at least 33.
Not on the formulae sheet: learn it.

This gives the total whether or not the angles are equal. Dividing it equally between vertices needs the extra information that the polygon is regular or equiangular.

Worked example 3: a total, non-calculator

Question. Work out the sum of the interior angles of a hexagon.

Plan. Count the triangles formed from one vertex.

n−2=6−2=4four trianglesS=4×180∘triangle angle sum=720∘multiply\begin{aligned} n-2&=6-2=4 &&\text{four triangles}\\ S&=4\times180^\circ &&\text{triangle angle sum}\\ &=720^\circ &&\text{multiply} \end{aligned}

Why this step? Six sides do not give six triangles from one vertex: the two sides meeting at that vertex account for the difference.

Answer. 720∘720^\circ.

Check. A pentagon totals 540∘540^\circ; adding one side adds one triangle, so a hexagon totals 540∘+180∘=720∘540^\circ+180^\circ=720^\circ.

The second example changes from finding a whole total to finding the missing part of that total.

Worked example 4: an irregular pentagon, non-calculator

Question. Four interior angles of a pentagon are 108∘108^\circ, 95∘95^\circ, 121∘121^\circ and 104∘104^\circ. Work out its fifth angle.

Plan. Find the pentagon's total, then subtract the four known angles.

S=(5−2)×180=540pentagon total108+95+121+104=428known anglesx=540−428=112remaining angle\begin{aligned} S&=(5-2)\times180=540 &&\text{pentagon total}\\ 108+95+121+104&=428 &&\text{known angles}\\ x&=540-428=112 &&\text{remaining angle} \end{aligned}

Why this step? The pentagon is not stated to be regular, so its angles cannot be assumed equal.

Answer. 112∘112^\circ.

Check. 108+95+121+104+112=540108+95+121+104+112=540.

The same derivation gives 1080∘1080^\circ for an octagon and 1440∘1440^\circ for a decagon. Naming a polygon tells you its side count; it does not tell you that it is regular.

Watch out

Wrong: every pentagon has angles of 108∘108^\circ. Right: every simple pentagon totals 540∘540^\circ, but its five individual angles need not match.

Keep the total and the individual angle separate in your working.

Continue this lesson.

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